D. S. Coad

Approximate bias calculations for sequentially designed experiments

A linear model is considered in which the design variables may be functions of previous responses and/or auxiliary randomisation. The model is observed successive times, where t is a stopping time, and interest lies in estimating the parameters of the model. Approximations are derived for the bias and variance of the maximum likelihood estimators of the parameters at time t. The derivations involve differentiating the fundamental identity of sequential analysis. The accuracy of the approximations is assessed by simulation for a multi-armed clinical trial model proposed by Coad (1995), two autoregressive models and the sequential design of Ford and Silvey (1980). Very weak expansions are used to justify the approximations.

Some results on estimation for two—stage clinical trials

A clinical trial setting is considered in which two treatments are available for a particular ailment. A two—stage trial is studied, in which patients are randomised equally in the first stage, and the better treatment at the end of this stage is used exclusively in the second stage. For exponential and Bernoulli responses, the exact bias and variance of the estimated treatment difference at the end of the trial are derived. Corresponding results for normal responses with unequal variances are also obtained, and the numerical accuracy of a normal approximation is investigated. The results indicate that the bias in estimation can be up to 25% when the size of the first stage is small, reducing to less than 7% for moderate first—stage sizes. For both exponential and Bernoulli responses, a normal approximation works well for moderate first—stage sizes, with the approximation for Bernoulli responses being slightly better.

Approximate confidence intervals after a sequential clinical trial comparing two exponential survival curves with censoring

Abstract A sequential test is considered for comparing two exponential survival curves with unknown failure rates θ1,θ2 > 0. The survival times are censored by both real time and an independent censoring variable. It is shown how very weak expansions for the bivariate version of the signed root transformation may be used to construct an approximate confidence interval for δ = log ( θ 1 θ 2 ) following the test. The accuracy of the method is illustrated by simulation results for several sequential tests and data-dependent allocation rules.

Sequential procedures for comparing several medical treatments

The use of sequential methods in clinical trials allows inferior treatments to be eliminated early. From an ethical point of view, the advantages are substantial. However, early stopping induces estimation bias and a deterioration in precision because of reduced sample sizes. This paper considers the problem of determining which of k ≥ 2 treatments with Bernoulli responses has the highest probability of success. Two sequential procedures are investigated and compared with a fixed—sample procedure. Various properties are derived and illustrated for the cases k =2,3 and 5. It is shown that the sequential procedures can achieve a pattern of error probabilities equivalent to the fixed—sample procedure for a much lower level of expected successes lost. Approximations for the bias and standard deviation of estimators of treatment differences are obtained by using results about the distribution of stopping times for a normal process.

Data-dependent allocation rules a comparative study of some for bernoulli data

A clinical trial setting is considered in which two treatments are available for a particular ailment. The response to treatment is Bernoulli with “success” or “failure” as the outcome. Several allocation rules which are designed to reduce the number of patients who receive the inferior treatment are compared with random allocation using simulation. Particular attention is paid to the bias and variance for estimation of the true treatment difference. The effect of time trends in the data is examined.

Corrected confidence intervals following a sequential adaptive clinical trial with binary responses

A sequential clinical trial model is considered in which two treatments are compared and the responses are binary. One of the properties of the model is that, for a given stopping boundary, the error probability is insensitive to the allocation rule. It is shown how a corrected confidence interval may be determined for the log odds ratio at the end of the trial. Simulation is used to assess the accuracy of the method for a selection of stopping boundaries and data-dependent allocation rules.

Sequential allocation rules for multi-armed clinical trials

A number of well-known sequential allocation rules are compared with a fixed-sample rule as procedures for determining which of k≥2 treatments with normal responses has the largest mean. For known variances, it is shown in the case k=2 how the parameters of the rules can be chosen so that they achieve a similar pattern of error probabilities. A simple modification of the rules is described to deal with the problem of unknown variences. Simulation indicates that the error probabilities for the modified rules are generally no larger than those for the fixed-sample rule. It is also shown that the rules extend naturally to the case of more than two treatments, with many of their important properties being preserved.

Sequential estimation for two-stage and three-stage clinical trials

Abstract A clinical trial setting is considered in which two treatments are available for a particular ailment. The responses to the treatments are normally distributed with unknown means and a common known variance. A two-stage trial and a three-stage trial are studied. In the two-stage trial, patients are randomised equally in the first stage, and the better treatment at the end of this stage is used exclusively in the second stage. The three-stage trial permits a second randomised stage before a single treatment is selected. For these designs, the exact bias and variance of the estimated treatment difference at the end of the trial are derived. These quantities are also derived when there are time trends in the data. Numerical results indicate that the presence of time trends can seriously bias the estimated treatment difference and can also lead to an increase in its variance.

CORRECTED CONFIDENCE INTERVALS FOR SECONDARY PARAMETERS FOLLOWING SEQUENTIAL TESTS

Corrected confidence intervals are developed for the mean of the second component of a bi- variate normal process when the first component is being monitored sequentially. This is accomplished by constructing a first approximation to a pivotal quantity, and then using very weak expansions to determine the correction terms. The asymptotic sampling distribution of the renormalised pivotal quantity is established in both the case where the covariance matrix is known and when it is unknown. The resulting approximations have a simple form and the results of a simulation study of two well- known sequential tests show that they are very accurate. The practical usefulness of the approach is illustrated by a real example of bivariate data. Detailed proofs of the main results are provided.

Sequential tests with covariates with an application to censored survival data

The problem of incorporating covariate information into sequential tests for deciding which of two treatments is superior is considered. It is assumed that the mean response depends on a number of covariates through an appropriate link function. Two large-sample sequential tests are described, and these tests are derived when the data are censored exponential survival times. Simulation results for random allocation indicate that the error probabilities for the new tests are generally no larger than those for tests derived in the absence of covariate information.